The applicability of the abstract results of sections  and  to the numerical approximation of Bayesian inverse problems encountered in practice is discussed in Section . We consider three representative model problems: the incompressible Euler equations (in 2d and 3d), the incompressible Navier-Stokes equations (in 3d) and a class of hyperbolic systems of conservation laws. For the incompressible Euler equations, we consider the numerical approximation by spectral schemes and verify the sufficient conditions for stability, compactness and consistency by a priori analysis for a class of priors in 2d. In 3d, the general stability and consistency properties continue to hold by the same a priori considerations; the compactness property holds under the additional assumption of a physically motivated bound on the structure functions. For the incompressible Navier-Stokes equations (in 3d), we prove the conditions for stability and compactness by a priori analysis, for numerical solutions obtained by spectral schemes. Finally, in the context of systems of hyperbolic conservation laws, we consider the entropy stable discretization by finite-volume schemes. Again, we prove stability for the BIP and filtering problems without any additional assumptions; compactness is obtained provided an average bound on the structure function holds. We point out that numerical evidence that the required bound on the structure function holds, has been demonstrated by numerical experiments for a number of initial data priors in [1]}, [2]}, [3]}, and is further motivated by physical considerations.
